Opinion 128:
Premature Abstraction, and Unrealistic Expectations, are the Root of All Evil:
Why are we Math Professors Doing such a TERRIBLE Job in Teaching Non-Math Majors (and Even
Math Majors) The Math That They Need

By Doron Zeilberger

Guru Don Knuth (who would celebrate 75-years in 4 days, Happy birthday Don!)
famously said, about computer programming (i.e. teaching computers to do stuff)
``Premature optimization is the root of all evil", and as Knuth taught us so well, we have
to start with the most naive approach, and then step-by-step make it better.

Analogously with students. There is such a great gap between what we assume that the
students know and can understand, and what they actually know and are able to understand.
When we teach calculus for example, we take it for granted that they know high-school algebra
and precalculus inside out. True, in order to enroll in Calc2, they were supposed to
know Calc1, which means that that they were supposed to know precalculus etc.
But they don't! Even those that get an A have a very shaky understanding of very
basic high-school algebra (that is much more important than calculus).
95 percent of even the A students get an A because they do the symbol-manipulations
correctly, and know how to reproduce the solutions in the review problems.
Those who are B and C students, are not as good fakers. This is like the
Turing Test, have a calculus student simulate understanding, but
very few actually understand what is going on.

And why should they? Calculus, and even high-school algebra, are very abstract.
For the prof., and even TA, it is all ``obvious'', so most of the instructors can't even
understand how it is "not possible to understand". The few professors (like myself) and TAs that
are actually (acutely!) aware of students' lack-of-understanding (and empathize with it)
are still bound by the demanding, largely obsolete, curriculum, teaching students
to do, mostly by rote, things that computer algebra systems can do so much better today.

The epitome of ignoring the abyss between the expected and actual
levels of students' understanding, and the great hypocrisy of our
system, here at Rutgers (and very possibly elsewhere), are extremely
challenging "workshops" that are fun for the TAs, but are way over the
heads of even the strongest students. What happens is that the TA
"guides" them to write a report, and those who can ape the guidelines
as faithfully as possible, get a good grade. Very few students actually
understand what is going on.

In fact, quite a few of them believe that 1/(a+b)=1/a+1/b, and log(a+b)=log(a)+log(b), and
that int(1/f(x),x)=log |f(x)|+C (for any f(x), not just x), etc. etc. They have no nonsense-correction instincts
that every future scientist and engineer should have. And it is not their fault!

It is not even our individual fault. It is our collective fault.
We should get a reality check, and gear our teaching level, and curriculum (making
full use of computers!) to the level that the students can understand and do well in.
Because of the great gap (I would call it ABYSS) between the actual level of the
students, and what we expect from them, often the average of the Final exam is
a failing grade. Since you can't fail everyone, professors start curving, and then
water boils at 0 degrees centigrade.

Curving should be prohibited, but the level of instruction should be geared to the
actual level of understanding, and background, of the students. And don't
be bashful about reviewing! Adding fractions, and algebraic simplifications, and
teaching them that sqrt(a+b) is NOT sqrt(a)+sqrt(b) are much more important than
partial fractions and integration by parts. Computers are such wonderful tools,
and one should make careful use of them, and use them to illustrate
the concepts.

It is amazing that many students, eventually, still come out OK, but this is
in spite of our dysfunctional system of instruction, not thanks to it,
and if we
would know to adjust the level of teaching to the actual level of
the students' background and abstraction level, and not to what they are "supposed to know and understand", they would be much better off!